The Spectral Radius of Graphs with No Intersecting Odd Cycles

The spectral radius of graphs with no intersecting odd cycles∗ Yongtao Li, Yuejian Peng† School of Mathematics, Hunan University Changsha, Hunan, 410082, P.R. China June 2, 2021 Abstract Let Hs,k be the graph defined by intersecting s triangles and k cycles of odd lengths at least five in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126–137], and Yuan [J. Graph Theory 89 (2018), no. 1, 26–39] determined independently the maximum number of edges in an n-vertex graph that does not contain Hs,k as a subgraph. In this paper, we determine the graphs of order n that attain the maximum spectral radius among all graphs containing no Hs,k for n large enough. Key words: Spectral radius; Intersecting odd cycles; Extremal graph; Stability mathod. 1 Introduction In this paper, we consider only simple and undirected graphs. Let G be a simple connected graph with vertex set V (G) = v , . , v and edge set E(G) = e ,...,e . Let d(v) or { 1 n} { 1 m} dG(v) be the degree of a vertex v in G. Let S be a set of vertices. We write dS(v) for the number of neighbors of v in the set S, that is, d (v)= N(v) S . And we denote by e(S) S | ∩ | the number of edges contained in S. The Turán number of a graph F is the maximum number of edges that may be in an n-vertex graph without a subgraph isomorphic to F , and it is usually denoted by ex(n, F ). We say that a graph G is F -free if it does not contain an isomorphic copy of F as a subgraph. arXiv:2106.00587v1 [math.CO] 19 May 2021 A graph on n vertices with no subgraph F and with ex(n, F ) edges is called an extremal graph for F and we denote by Ex(n, F ) the set of all extremal graphs on n vertices for F . It is a cornerstone of extremal graph theory to understand ex(n, F ) and Ex(n, F ) for various graphs F ; see [23, 27, 39] for surveys. In 1941, Turán [40] posed the natural question of determining ex(n, K ) for r 2. r+1 ≥ Let Tr(n) denote the complete r-partite graph on n vertices where its part sizes are as equal as possible. Turán [40] (also see [5, p. 294]) extended a result of Mantel [29] and obtained that if G is an n-vertex graph containing no K , then e(G) e(T (n)), equality holds r+1 ≤ r if and only if G = Tr(n). There are many extension and generalization on Turán’s result. The problem of determining ex(n, F ) is usually called the Turán-type extremal problem. ∗ E-mail addresses: [email protected] (Y. Li), [email protected] (Y. Peng, corresponding author). 1 The most celebrated extension always attributes to a result of Erd˝os, Stone and Simonovits [14, 13], which states that 1 n2 ex(n, F )= 1 + o(1) , (1) − χ(H) 1 2 − where χ(F ) is the vertex-chromatic number of H. This provides good asymptotic estimates for the extremal numbers of non-bipartite graphs. However, for bipartite graphs, where χ(F ) = 2, it only gives the bound ex(n, F ) = o(n2). Although there have been numerous attempts on finding better bounds of ex(n, F ) for various bipartite graphs F , we know very little in this case. The history of such a case began in 1954 with the Kövari-Sós-Turán theorem [28], which states that if Ks,t is the complete bipartite graph with vertex classes of size s t, then ex(n, K ) = O(n2−1/t); see [19, 20] for more details. In particular, we ≥ s,t refer the interested reader to the comprehensive survey by Füredi and Simonovits [23]. 1.1 History and background In this section, we shall review the exact values of ex(n, F ) for some special graphs F , instead of the asymptotic estimation. A graph on 2k + 1 vertices consisting of k triangles which intersect in exactly one common vertex is called a k-fan (also known as the friendship graph) and denoted by Fk. Since χ(Fk) = 3, the Erd˝os-Stone-Simonovits theorem in (1) 2 2 implies that ex(n, Fk)= n /4+ o(n ). In 1995, Erd˝os et al. [15] proved the following exact result. Theorem 1.1. [15] For every k 1, and for every n 50k2, ≥ ≥ n2 k2 k, if k is odd, ex(n, F )= + − k 4 k2 3 k, if k is even. − 2 The extremal graphs of Theorem 1.1 are as follows. For odd k (where n 4k 1), ≥ − the extremal graph is uniquely constructed by taking a complete bipartite graph with color classes of size n and n and embedding two vertex disjoint copies of K in one side. For ⌈ 2 ⌉ ⌊ 2 ⌋ k even k (where now n 4k 3), the extremal graph in not unique, and each extremal graph ≥ − is constructed by taking a balanced complete bipartite graph and embedding a graph with 2k 1 vertices, k2 3 k edges with maximum degree k 1 in one side. − − 2 − Let Ck,q be the graph consisting of k cycles of length q which intersect exactly in one common vertex. Clearly, when we set q = 3, then Ck,3 is just the k-fan graph; see Theorem 1.1. When q is an odd integer, we can see that χ(Ck,q) = 3, the Erd˝os-Stone-Simonovits 2 2 theorem also implies that ex(n,Ck,q) = n /4 + o(n ). In 2016, Hou, Qiu and Liu [25] determined exactly the extremal number for C with k 1 and odd integer q 5. k,q ≥ ≥ Theorem 1.2. [25] For an integer k 1 and an odd integer q 5, there exists n (k,q) ≥ ≥ 0 such that for all n n (k,q), we have ≥ 0 n2 ex(n,C )= + (k 1)2. k,q 4 − Moreover, an extremal graph must be a Turán graph T2(n) with a Kk−1,k−1 embedding into one class. 2 We remark here that when q is even, then Ck,q is a bipartite graph where every vertex in one of its parts has degree at most 2. For such a sparse bipartite graph, a classical result 3/2 of Füredi [18] or Alon, Krivelevich and Sudakov [2] implies that ex(n,Ck,q) = O(n ). Recently, a breakthrough result of Conlon, Lee and Janzer [10, 11] shows that for even q 6 and k 1, we have ex(n,C ) = O(n3/2−δ) for some δ = δ(k,q) > 0. It is a ≥ ≥ k,q challenging problem to determine the value δ(k,q). For instance, the special case k = 1, this problem reduces to determine the extremal number for even cycle. Next, we shall introduce a unified extension of both Theorem 1.1 and Theorem 1.2. Let s, k be integers and let Hs,k be a graph consisting of s triangles and k cycles of odd lengths at least 5 which intersect in exactly one common vertex. The graph Hs,k is also known as the flower graph with s + k petals. We remark here that the k odd cycles can have different length. Clearly, when k = 0, then Hs,0 = Fs, the s-fan graph; see Theorem 1.1. In addition, when s = 0 and the lengths of odd cycles are all equal to q, then H0,k = Ck,q; see Theorem 1.2. In 2018, Hou, Qiu and Liu [26] and Yuan [42] independently determined the extremal number of H for s 0 and k 1. Let be the family of graphs with each member s,k ≥ ≥ Fn,s,k being a Turán graph T2(n) with a graph H embedded in one partite set, where K , if (s, k) = (3, 1), H = s+k−1,s+k−1 6 (K3,3 or 3K3, if (s, k) = (3, 1), where 3K3 is the union of three disjoint triangles. Theorem 1.3. [26, 42] For two integers s 0, k 1, there exists n (s, k) such that for all ≥ ≥ 0 n n0(s, k), we have ≥ n2 ex(n,H )= + (s + k 1)2. s,k 4 − Moreover, the only extremal graphs for H are members of . s,k Fn,s,k 1.2 Spectral extremal problem Let G be a simple graph on n vertices. The adjacency matrix of G is defined as A(G) = (aij)n×n with aij = 1 if two vertices vi and vj are adjacent in G, and aij = 0 otherwise. We say that G has eigenvalues λ1, . , λn if these values are eigenvalues of the adjacency matrix A(G). Let λ(G) be the maximum value in absolute among the eigenvalues of G, which is known as the spectral radius of graph G, that is, λ(G) = max λ : λ is an eigenvalue of G . {| | } By the Perron–Frobenius Theorem [24, p. 534], the spectral radius of a graph G is actually the largest eigenvalue of G since the adjacency matrix A(G) is nonnegative. The spectral radius of a graph sometimes can give some informations about the structure of graphs. For example, it is well-known [4, p. 34] that the average degree of G is at most λ(G), which is at most the maximum degree of G.

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